Problem 26 - Precalculus

Question

Use the properties of logarithms to write the expression as a single logarithm. $$ \ln (x)+\frac{1}{2} $$

Step-by-Step Solution

Answer

The single logarithm is \( \ln(x \sqrt{e}) \).

Step 1: Recall the property of logarithms

One important property of logarithms you should know is that the expression \( a \, \ln(b) \) can be rewritten using the power rule for logarithms as \( \ln(b^a) \). This will be useful for simplifying the fraction in the expression.

Step 2: Apply the power rule of logarithms

Given the expression \( \ln(x) + \frac{1}{2} \), apply the power rule to transform \( \frac{1}{2} \) into a logarithmic form. Rewrite \( \frac{1}{2} \) as \( \ln(e^{1/2}) \). This works because the natural logarithm of \( e \) to any power is simply that exponent.

Step 3: Combine the logarithms

Now that the expression is \( \ln(x) + \ln(e^{1/2}) \), apply the property of logarithms that \( \ln(a) + \ln(b) = \ln(a \cdot b) \). This simplifies our expression to \( \ln(x \cdot e^{1/2}) \).

Step 4: Simplify the expression

To simplify \( \ln(x \cdot e^{1/2}) \), recognize that \( e^{1/2} \) is equivalent to \( \sqrt{e} \). Therefore, the expression can be rewritten as \( \ln(x \sqrt{e}) \).

Key Concepts

Properties of Logarithms

Logarithms have several fundamental properties that allow us to manipulate and simplify expressions. These properties are crucial for tackling various mathematical problems involving logarithms. Here are the key properties:

• Product Property: The logarithm of a product is the sum of the logarithms. In mathematical terms: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \).
• Power Property: The logarithm of a power is the exponent times the logarithm of the base: \( \ln(b^a) = a \cdot \ln(b) \).

Each of these properties enables us to decompose complex logarithmic expressions into more manageable parts or to combine several logarithms into a single expression, which is particularly helpful when solving equations or expressing logarithmic identities.

Power Rule of Logarithms

The power rule of logarithms is one of the pivotal tools for simplifying logarithmic expressions, especially those involving exponents. This rule states that the logarithm of an exponential expression is the exponent times the logarithm of the base. For example, this can be written mathematically as:

• \( \ln(b^a) = a \cdot \ln(b) \)

This rule allows us to pull down exponents as coefficients, simplifying expressions significantly. For instance, if you encounter an expression like \( \ln(x^2) \), applying the power rule transforms it into \( 2 \cdot \ln(x) \). In the exercise example, the power rule helps to convert the fraction \( \frac{1}{2} \) into a logarithmic form, which is essential to combine it with another logarithmic expression.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \) is approximately 2.71828, a mathematical constant known as Euler's number. It is widely used in calculus and analysis due to its unique properties when dealing with growth processes and complex calculations.Natural logarithms have several interesting features:

• The natural logarithm of \( e \) is 1: \( \ln(e) = 1 \).
• It's commonly used in situations involving continuous growth or decay, such as in population models or radioactive decay.
• Natural logarithms simplify complex relationships involving exponential growth, making them invaluable in fields like physics and finance.

With the natural logarithm, expressions involving \( e \) are often more straightforward to handle, especially when combined with properties like the power rule, as seen in the exercise example.

Single Logarithm

Combining logarithms into a single expression simplifies mathematical operations and makes understanding the relations between quantities easier. This is done by utilizing the properties of logarithms. For instance, if you have multiple logarithmic terms like \( \ln(x) + \ln(y) \), you can apply the product property to merge them into a single logarithm: \( \ln(x \cdot y) \). This consolidation process reduces computational complexity and aids in solving logarithmic equations more swiftly.In the provided exercise, the transformation of \( \ln(x) + \ln(e^{1/2}) \) into \( \ln(x \cdot e^{1/2}) \) is an example of writing an expression as a single logarithm. This final form is concise and clear, showing the relationship between \( x \) and \( e^{1/2} \) in a unified way, effectively demonstrating how multiple logarithms can reflect a simpler concept when combined.